Assume that B is a Boolean algebra with operations+ and

Chapter 6, Problem 6.165

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Assume that B is a Boolean algebra with operations+ and . Prove each statement without using any parts of Theorem 6.4.1 unless they have already been proved. You may use any part of the denition of a Boolean algebra and the results of previous exercises, however. Let S ={ 0,1}, and dene operations + and on S by thefollowing tables: + 01 0 01 1 11 01 0 00 1 01 a. Show that the elements of S satisfy the following properties: (i) the commutative law for+ (ii) the commutative law for (iii) the associative law for+ (iv) the associative law for (v)H the distributive law for+over (vi) the distributive law forover+b.H Show that 0 is an identity element for+and that 1 is an identity element for. c. Dene 0=1 and1=0. Show that for all a in S, a+a =1 andaa =0. It follows from parts (a)(c) that S is a Boolean algebra with the operations+and.